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Theorem axssetprim 4092
 Description: ax-sset 4082 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)
Assertion
Ref Expression
axssetprim xyz(a(b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))) a x) ↔ e(e ye z))
Distinct variable groups:   a,b   x,a,y,z   b,c   b,d,y,z   y,c   y,d,z   x,e,y,z

Proof of Theorem axssetprim
StepHypRef Expression
1 ax-sset 4082 . 2 xyz(⟪y, z xe(e ye z))
2 df-clel 2349 . . . . . 6 (⟪y, z xa(a = ⟪y, z a x))
3 axprimlem2 4089 . . . . . . . 8 (a = ⟪y, z⟫ ↔ b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))))
43anbi1i 676 . . . . . . 7 ((a = ⟪y, z a x) ↔ (b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))) a x))
54exbii 1582 . . . . . 6 (a(a = ⟪y, z a x) ↔ a(b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))) a x))
62, 5bitri 240 . . . . 5 (⟪y, z xa(b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))) a x))
76bibi1i 305 . . . 4 ((⟪y, z xe(e ye z)) ↔ (a(b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))) a x) ↔ e(e ye z)))
872albii 1567 . . 3 (yz(⟪y, z xe(e ye z)) ↔ yz(a(b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))) a x) ↔ e(e ye z)))
98exbii 1582 . 2 (xyz(⟪y, z xe(e ye z)) ↔ xyz(a(b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))) a x) ↔ e(e ye z)))
101, 9mpbi 199 1 xyz(a(b(b a ↔ (c(c bc = y) d(d b ↔ (d = y d = z)))) a x) ↔ e(e ye z))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟪copk 4057 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-sset 4082 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-opk 4058 This theorem is referenced by: (None)
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